# De Morgan's Laws (Set Theory)/Set Complement/Complement of Union/Proof 2

## Theorem

$\overline {T_1 \cup T_2} = \overline T_1 \cap \overline T_2$

## Proof

 $\ds$  $\ds x \in \overline {T_1 \cup T_2}$ $\ds$ $\leadstoandfrom$ $\ds x \notin \paren {T_1 \cup T_2}$ Definition of Set Complement $\ds$ $\leadstoandfrom$ $\ds \neg \paren {x \in T_1 \lor x \in T_2}$ Definition of Set Union $\ds$ $\leadstoandfrom$ $\ds \neg \paren {x \in T_1} \land \neg \paren {x \in T_2}$ De Morgan's Laws (Logic): Conjunction of Negations $\ds$ $\leadstoandfrom$ $\ds x \in \overline {T_1} \land x \in \overline {T_2}$ Definition of Set Complement $\ds$ $\leadstoandfrom$ $\ds x \in \overline {T_1} \cap \overline {T_2}$

By definition of set equality:

$\overline {T_1 \cup T_2} = \overline {T_1} \cap \overline {T_2}$

$\blacksquare$